Constraints on flavored 2d CFT partition functions

Journal of High Energy Physics, Feb 2018

Ethan Dyer, A. Liam Fitzpatrick, Yuan Xin

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP02%282018%29148.pdf

Constraints on flavored 2d CFT partition functions

HJE Constraints on avored 2d CFT partition functions Ethan Dyer 0 1 2 4 5 A. Liam Fitzpatrick 0 1 2 3 Yuan Xin 0 1 2 3 0 Charles Street, Baltimore, MD 21218 , U.S.A 1 Commonwealth Avenue , Boston, MA 02215 , U.S.A 2 Via Pueblo , Stanford, CA 94305 , U.S.A 3 Physics Department, Boston University 4 Department of Physics and Astronomy, Johns Hopkins University 5 Stanford Institute for Theoretical Physics We study the implications of modular invariance on 2d CFT partition functions with abelian or non-abelian currents when chemical potentials for the charges are turned on, i.e. when the partition functions are avored". We begin with a new proof of the transformation law for the modular transformation of such partition functions. Then we proceed to apply modular bootstrap techniques to constrain the spectrum of charged states in the theory. We improve previous upper bounds on the state with the greatest mass-tocharge" ratio in such theories, as well as upper bounds on the weight of the lightest charged state and the charge of the weakest charged state in the theory. We apply the extremal functional method to theories that saturate such bounds, and in several cases we nd the resulting prediction for the occupation numbers are precisely integers. Because such theories sometimes do not saturate a bound on the full space of states but do saturate a bound in the neutral sector of states, we nd that adding method to solve for some partition functions that would not be accessible to it otherwise. Conformal and W Symmetry; Global Symmetries - Semi-de nite programming with continuous charge Q Bound on dimension of lightest charged state 4.5.1 4.5.2 Bound on charge-to-mass ratio Bound on lowest charge Extremal functional analysis and Q approach | revisiting the E8 lattice 1 Introduction 2 3 4 2.1 2.2 3.1 3.2 4.1 4.2 4.3 4.4 4.5 5.1 5.2 Partition function transformation and background gauge elds Modular transformation and the ground state energy Non-abelian current transformation Modular bootstrap with chemical potentials Basic setup Semide nite projective functionals and the extremal method Abelian bounds 5 Non-abelian bounds Bounds on gaps in operator dimensions Extremal functional analysis at c = 3 5.2.1 5.2.2 Spin-independent analysis Spin-dependent analysis 5.3 Constraints on representation content 6 Discussion and future directions A Path integral modular transformation B A \systematic" treatment to multivariate problems B.1 Multivariate positive de nite functionals B.2 Multivariate problems and SDPB C k = 2; SU(2) analysis { 1 { Introduction Modular invariance is a powerful tool for studying two-dimensional Conformal Field Theories (CFTs). It is also a special case of crossing symmetry of CFT correlation functions [1], so aside from its intrinsic interest it is useful as a simpler setting in which to explore many conformal bootstrap ideas and techniques [2{4]. A particularly appealing generalization of the conformal bootstrap equations is to consider correlation functions in the presence of nonlocal operators, since this enlarges the set of CFT data that can be studied. In general, including nonlocal operators is a di cult problem, since their behavior under conformal transformations may be quite complicated. However, one case where the problem remains tractable is when we consider modular invariance in the presence of a chemical potential in 2d CFTs. A chemical potential corresponds to inserting the nonlocal operator yJ0 e2 izJ0 into the partition function, Z( ; z) tr qL0 2c4 qL0 2c4 yJ0 ; where J0 is the zero mode of a conserved current and L0; L0 are Virasoro generators. The resulting partition function is no longer modular invariant, but nevertheless has a wellde ned and theory-independent transformation law [5]: Z a + b cz ; 1We emphasize that we do not assume supersymmetry; additional techniques are available to proof the transformation law for the elliptic genus in the case of supersymmetric theories, see e.g. [9]. { 2 { We also improve the bound on the smallest \mass-to-charge" ratio in the theory. These bounds are qualitatively related to the Weak Gravity Conjecture (WGC), though gauge elds in the gravity duals are Chern-Simons elds rather than Maxwell elds. Provocatively, we nd numerical evidence for a bound on the mass-to-charge ratio that scales at large c as pc, consistent with the bulk gravity expectation. This is stronger than the bounds in [6], which scale as c. The improvement again comes from increasing the number of derivatives of the characters used in the analysis. Then, we discuss the bound on the charge gap Q . Without any further assumptions, the numerical bound of charge gap is always Q = 1 for all c. We study two examples of c = 2 and 8 by turning on both a gap in dimension and in charge Q . There are kinks are able to reproduce the full low lying spectrum, including charge assignments. Non-abelian bounds. When the symmetry current J a is non-abelian, it is more appropriate to consider bounds on the dimensions of di erent representations in the theory. We will mainly focus for speci city on the case where the gauge group G is SU(2) and the level k is 1, though our methods easily generalize to any algebra and level; the main advantage of k = 1; G = SU(2) is that convergence is fastest here, so our numerical results are most precise. We rst obtain bounds on the gap to all non-vacuum states in non-abelian theories. As the extended symmetry imposes additional relations on the spectrum, one may have hoped for stronger bounds. The results, however, are similar to those found in the abelian, or even non- avored case. In particular, at large c we nd a bound of the form, = c 2The partition functions found are not strictly modular invariant, but invariant under a subgroup generated by S and T n for n = 2 or n = 4. { 3 { the gap. function for this case, but rather that fact that searching for constraints in a representation dependent manner yields structure hidden to a avor-blind analysis. This means that the extremal functional analysis allows one to \discover" a larger class of partition functions when avored information is included than when it is not. Moreover, uncovering avored information can potentially split the degeneracy between theories with the same spectrum and therefore the same partition function, allowing us to address the age-old question of whether one can \taste the shape of a drum." Finally, we continue to re ne our representation dependent analysis. For the case of SU(2)1 we prove analytically that the theory either contains all representation, or the partition function splits into a product of the diagonal Sugawara partition function and a neutral, modular invariant partition function. After this work was completed, the paper [11] appeared on arXiv also considering modular bootstrap constraints on theories with conserved currents, though the analysis there did not use the avored partition function. 2 Partition function transformation and background gauge elds In this section, we will present an argument for the transformation law (1.2) based on the e ective action obtained upon integrating out the CFT in the presence of a background gauge eld A . Previous treatments have pointed out that in the present context there are two di erent notions of the partition function that are natural. One of these is the canonical partition function Z( ; z) de ned by (1.1). Following [5], we will refer to an alternate de nition as the \path integral" ZPI( ; z): Under z0 = c z+d ; 0 = ac ++db , the factor B is easily seen to transform as First, let us discuss in more detail how to de ne the \path integral" function Z( ; z), what ambiguities are allowed in this de nition, and why they do not a ect the transformation law (1.2). In order to be invariant under modular transformations, we will need to de ne the path integral to be invariant under di eomorphisms and rigid rescalings w ! d e S [ ] = d 0e S 0 [ 0]: Here, are all the elds of the CFT. As we review in appendix A, these two symmetries are su cient to imply that the path integral de ned as an integral over this measure, is invariant under modular transformations: Z ZPI 1w: (2.3) (2.4) (2.5) (2.6) ; a + b cz J . However, it is easily seen to be both Weyl invariant and di eomorphism invariant, and is invariant under modular transformations. So its coe cient is irrelevant for our purposes, and from now on we will neglect such terms without loss of generality. Now, the next question is how do we relate the \path integral" ZPI to the partition function Z? The key point is that turning on a background eld A not only turns on a chemical potential, but it can also shift the ground state energy, since at xed such a shift a ects only the overall normalization of the path integral. In the example of the free boson, this energy shift is seen explicitly by doing a Legendre transform, but we can see it in full generality by considering the e ective action for A . To see the shift, it is su cient to calculate the ground state energy, so we can take the limit of the torus where = i 2 ; 1. In this limit, the torus becomes a cylinder, and the e ective action is conformally related to that in at space, where it is universal and known in closed form. Including the action for a background metric as well, we can write log Z = Z d x 2 p g c 48 R 1R + k 8 F 1 F : (2.7) SWZ = 24 c R d2xp g Because of the inverse Laplacians, the mapping to the cylinder is a bit subtle. For the metric contribution, it is easiest to work with the Wess-Zumino anomaly action directly, R + (@ )2 , and take (w) = w+w, which reproduces the standard { 5 { Combining the above with a symmetric combination from Aw, we put everything together to obtain the ground state energy: E0 = lim !1 c 12 1 log ZPI = + E; this factor is universally the contribution to the partition function from the shift in the ground state energy due to the background gauge eld. Summarizing, the canonical partition function Z( ; z) in (1.1) is de ned to have a ground state energy 1c2 . However, any path integral over the torus using a regulator that preserves di eomorphisms and rescalings will have a ground state energy equal to k(A2w + A2w), plus possible terms that do not a ect the modular transformation of Z( ; z). c 12 2.2 Non-abelian current transformation The generalization to the case of a non-abelian is straightforward, and can be made as follows. Unlike in the abelian case, the e ective action is not quadratic. However, we can write it formally as the sum over all connected diagrams: WA[Aa ] = 1 X the d2w0 integral, producing just a factor of the volume 2 of the torus. Passing to t; coordinates: 3We performed this integration as follows. First, shift w ! w + w0 to eliminate w0 and immediately do WA[A ] ! (2 i)2 k Z d2wd2w0 1c2 from the Schwarzian derivative. By contrast, the gauge eld term in (2.7) is invariant under Weyl transformations, and its contribution to the ground state energy just comes from evaluating the non-local term on the cylinder. To avoid ambiguities associated with the inverse Laplacian, it is clearest to use the fact that the e ective action is the generating functional for the J correlators, so we know that we can equivalently write the gauge eld part of log Z as On the plane, hJ w(w)J w(w0)i = (w kw0)2 . Mapping to the cylinder and taking Aw to be As before, we want to set Aaw; Aaw to be constant on the cylinder and integrate over d2wi. For the part quadratic in A, the computation proceeds just as in the abelian case, we simply have an extra index for the di erent components of Jwa . The background eld couples as A's contribution to the ground state energy is which transforms under modular transformations as 2 i Z Aa J a; Aaw = i z a means `up to non-singular terms'. The kwa2b piece manifestly generates disconnected diagrams - it produces the two-point function times the (n 2)-point function - so it does not contribute to the e ective action for higher-point correlators. But, since we multiply the correlator by Aaw in the e ective action, the f abc term also gives no contribution for constant Aaw: AawJ a(w)AbwJ b(0) w2 kA2w + AawAbwf abc J c(0) = w kA2w w2 since AaAbf abc = 0. Therefore only the two-point functions contribute. That leaves the contribution from the higher-point functions. We can always write these in terms of lower-point function by using the recursive formula c(0) = c csug; csug = kjGj : k + ehG { 7 { 3 3.1 Basic setup Modular bootstrap with chemical potentials In all the cases we consider, we will assume the presence of a conserved current J a in the theory. In general, it is convenient to separate the stress tensor T of the theory into a Sugawara stress tensor piece and a residual piece: T (0) T T sug; T sug = 1=2 k + ehG a=1 jGj X : J aJ a :; where ehG is the dual Coxter number, because the modes of T (0) commute with the modes of J a. Furthermore among themselves they form a Virasoro algebra with central charge Similarly, we can separate the Virasoro generators Ln = L(n0) + Lsnug and the weights h = h(0) + hsug into a part that comes from T (0) and a part that comes from T sug. For most representations, the distinction between T and T (0) will not make much di erence, since the partition function just counts states at each level. However, for the special cases with shortening conditions, some descendants becomes null and do not contribute to the partition function, and this is easier to see using the modes of T (0). The characters of the Kac-moody algebra X ;k( ; z) = TrV ;k q 0 are constructed by acting modes of J a on some highest weight state which has weight . Hmeerter,y,HH0 0is=thJe0 vaencdtorthoef cChaarrtaacntegresnfeorrataorgsenoefrticheparilmgeabrrya.arIen stihmepclayseq ocfsua2g4n 1aeb2eliizaQn=sy(m)-. In the case of a non-abelian symmetry, the characters are more complicated. Some descendants of such a highest weight state may be null so it is non-trivial to write down its form. However, for the purpose of the modular bootstrap, the only property of such characters we use is that the characters transform covariantly where the matrix S depends on the symmetry group and level k. These characters do not include the modes of T (0) yet. Since the algebra generated by modes of J a is completely orthogonal to that generated by modes of T (0), the character generated by the full extended algebra simply factorizes into a Kac-Moody character and a Virasoro character Like the simple Virasoro character, the character is di erent if the primary saturates the unitarity bound: X ;k;h( ; z) = X ;k( ; z)Xh(0) ( ) : Xh(0) ( ) = < 8 > qh(0) c(02)4 1 The same goes for the anti-holomorphic part. The full partition function is X ; ;h;h d ; ;h;hX ;k( ; z)X ;k( ; z)Xh(0) ( )Xh(0) ( ) : In the above equation the means the representation of the anti-holomorphic part. When we are dealing with a non-abelian symmetry, it will be convenient to de ne a matrix M ; whose components are the coe cients of the contributions to the partition function from the di erent representations: h;h X ; M ( ; ) ; = X d ; ;h;hXh(0) ( )Xh(0) ( ) { 8 { (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) Modular transformations on the partition function translates into a speci c modular transformation of the matrix M ; . To see this transformation law, simply separate out the transformation law of Z into its irrep constituents: 0 = Z 1 ; where we have used the transformation rule, (3.4), and the de nition (3.8). Stripping o the characters, the above crossing equation is equivalent to a crossing equation for the matrix M ( ; ) ; X ;k( ; z)X ;k( ; z) (3.9) 0; 0 2 z 2 = X X Sk0 M ; X ;k 1 ; 1 X ;k( ; z)X ;k( ; z) ; 0 = M ( ; ) ; ST; 0 M (3.10) (3.11) (3.12) For the constraints on theories with non-abelian currents, equation (3.10) is the form of the constraint that we will use. For each bootstrap question we will input the symmetry group and level k. 3.2 Semide nite projective functionals and the extremal method To be self-contained, we will brie y review linear and semide nite programming methods as applied to the modular bootstrap; for more thorough reviews and some examples of applications, see e.g. [4, 10, 13{18], or [19{25] for reviews and some of the original papers developing methods in the standard bootstrap that we will adopt directly. The starting point is equation (3.9), which can be written 0 = d ; ;h;h F ; ;h;h 0; 0 ( ; ); X h;h; 0; 0 h 0 F ; ;h;h 0; 0 ( ; ) 0 Xh(0) ( )Xh(0) ( ) S 0 S 0; Xh(0) ( 1= )Xh(0) ( 1= ) : i The occupation numbers d ; ;h;h are all non-negative, and include in particular the vacuum dvac = 1. One is generally interested in proving that there exist states in the theory with various properties, for instance that there exists a state with < max for some value of max. Let us abstractly call a choice of such properties \P ". Then, one can prove that there is at least one state in the theory with properties P as long as one can nd a linear functional that maps the characters to real numbers such that it is positive on the vacuum and also positive on all states not satisfying P . In equations, (vac) = 1; and (F ; ;h;h) 0 unless ( ; ; h; h) satis es P: (3.13) { 9 { The normalization (vac) = 1 is conventional. If such a linear functional exists, then there must be a state in the theory with the properties P , otherwise acting on equation (3.11) would imply 0 1. Usually we will be interested in not just one choice of P but a continuous family of choices Ps parameterized by a continuous variable (or variables) s. Typically, s will be something like the bound max in the example above, so that as one decreases s the set of states with property Ps grows and therefore the set of linear functionals that are positive on all such states shrinks. Critical values s of s where the set of such linear functionals vanishes are especially interesting: aside from giving the best possible bounds, at these points one can use the \extremal functional method" [19] to determine all of the occupation numbers d ; ;h;h. The basic idea behind this is that for any s, the space of functions F ; ;h;h spanned by states satisfying Ps is a polytope where Fvac is inside the polytope for s < s and outside the polytope for s > s . At exactly s , Fvac passes through one of the faces of the polytope, so there is a unique positive semide nite linear combination of the states satisfying Ps that cancels the contribution from the vacuum in (3.11). In practice, we have to work with nite-dimensional projections of the full space of functions F ; ;h;h, but one optimistically expects to converge to a unique solution as the dimensionality of the projected space increases. We will encounter some exceptions that we will discuss as we come to them. 4 Abelian bounds In this section, we perform a systematic numerical analysis on the bounds on the gap in dimensions and charges, as well as on the smallest charge-to-mass ratio allowed in a theory with a U( 1 ) current. 4.1 Semi-de nite programming with continuous charge Q For the abelian case, for simplicity we will not use the full Kac-Moody characters, but rather just the Virasoro characters h(q): Z( ; z) = X h;h;Q;Q dQ;Q;h;hyQyQ h(q) h(q): where q = e2 i , q = e 2 i , y = e2 iz, y = e 2 iz. z = 0. we reduce the partition function as We will consider left-right symmetric theories with c = c, and for simplicity we set 1 As in [4], we reduce the characters using the S invariant factor j j 2 j ( )j2. Furthermore Z^( ; z) so that Z^( ; z) is invariant under z . The characters are reduced into ^0(q) = e 2 i z2 1 (4.1) (4.2) (4.3) We consider linear functionals of the form where the change of variable under S transformations.5 We arrive at a two variable functional X m+n+k=2 odd; k even k t=t=w=0 ; (4.4) = iet and z = we 2t is made so that t 7! t and w2 7! w2 l where we assume the spectrum is parity symmetric. Part of the challenge of the abelian analysis is that we do not assume charge quantization, i.e. technically we allow the gauge group to be R instead of U( 1 ), which means that we have to deal with not just one but two continuous parameters, and Q. This complicates the application of positive semi-de nite approaches, since these are based on constructing positive functionals of the characters and in general the space of such functionals is more complicated for multiple variables than for a single variable. In particular, for a single variable, positive semi-de nite functionals can be written without loss of generality as a sum of squares plus a linear term times a sum of squares. For multiple variables, such a parameterization is no longer completely general. One way to deal with this issue is simply to discretize in, say, Q, but we nd that such an approach becomes di cult to implement in practice since the discretization needs to become very ne to prevent the numeric search from picking functionals that become negative in between the discretization points. The approach we take is instead to limit the search space to functionals that are still a sum of squares plus a linear term times sums of squares. In the limit of very high order polynomials, one might expect that such functionals can approximate the extremal functionals arbitrarily well. In any case, while such functionals might not give the best possible bounds, they nevertheless produce valid bounds. Even restricting to polynomial functionals, there remains a practical problem of how to implement the search over such functionals using available software for semi-de nite programming analyses. In appendix B, we discuss how to massage this problem into an appropriate form for use with SDPB. 4.2 Bound on dimension of lightest charged state With the avored partition function we can bound the dimension of lightest charged state in any theory with a U( 1 ). The bound for di erent c is summerized in table. 1. We extrapolate the bound values to nD ! 1 using a linear function of n1D similar to what is done in [10] for c 100 where the convergence of the bound values is signi cant. Then we 5In this expression for , we have not used any z derivatives and so do not use information about the perform one analysis where we keep z derivatives to demonstrate this point. 2:40 2:00 1:90 1:85 1:80 1:79 1:79 1:78 1:78 1:77 6:40 5:60 5:00 4:90 4:80 4:70 4:60 4:55 4:53 4:50 19:2 16:0 15:6 15:2 14:8 14:4 14:0 13:8 13:6 13:5 102: 51:2 49:6 48:8 47:2 46:4 46:0 45:2 44:8 44:4 179:2 166:4 160:0 158:4 155:2 153:6 152:0 150:4 148:8 148:0 as a function of c and the number nD of derivatives used in the bootstrap functionals. = 2i extrapolate the bounds to nD ! 1 and c ! 1 by tting the nite nD and c results to a linear function of 1c and 1=nD and extrapolating, as shown in gure 1. In this test we take with real to avoid the complication of spin. It is not understood a priori why the form of this t works, but empirically it agrees well with the data at large c and nD. Similarly to the results of [10], extrapolating in nD and then c provides a parametrically stronger bound than the nite nD analysis. = c (4.6) HJEP02( 18 )4 metry. The extrapolated gaps at nD ! 1 with the trend line. This bound (4.6) is similar to the bounds on the non-charged state found in [4, 10], though quantitatively di erent. The bound in [4] is parametrically weaker, which is not surprising since that analysis did not use the spins of the characters, and did not perform any extrapolation in the number of derivatives. The bound in [10] is more analogous since spins and extrapolations were used; the result there is very slightly stronger ( 9) than (4.6) for the charged spectrum. 4.3 Bound on charge-to-mass ratio In this section, we will present results that there must be a state in the theory with a charge-to-mass ratio r Qc 12 = Q 8GN m ; above some critical value r , whose value we will determine numerically.6 We try to nd the linear functional that7 (4.7) (4.8) (4.9) HJEP02( 18 )4 (vac) jQj Q 12r c : X m+k=2 odd; k even We drop the spin index l by only taking functionals of the form 6The value of r increases as the number nD of derivatives used increases and the numeric accuracy improves, though we emphasize that even for low number of derivatives the values of r are a valid bound 7We also impose a dimension cuto nite dimension. Di erent dimension cuto s do not result in signi cantly di erent functionals and bounds. It just helps the algorithm to nd a functional that only is negative at nite . Q / N4 m G 8 2 0 comparison. The extrapolation (\nD = 1" points) and error bars are computed by performing a t as a function of nD and extrapolating to nD ! 1 as described in the text. For the functional to be positive in a bounded region we make a change of variables of the form ~ jQj Q2 = Q~2Q2 Q~2 + 1 : (4.10) 0 means jQj Q , inspired by [10]. optimal bound on r result in [6] was 8GN m Q First, we show in gure 2 the bound on r as a function of c. By inspection, one can see that the larger c is, the longer it takes for the bounds to converge. To see how the bound depends on the number nD of derivatives in more detail, in gure 3 we focus on a speci c value of c, c = 105 and show the resulting bound on r as a function of the number nD of derivatives allows in the functional . The best t as power law suggests that the might be signi cantly better, i.e. (r ) 1 1. For comparison, the = (r ) 1 < 4 p = 7:1. In [6], it was also shown that even with a small number nD of derivatives, one could obtain a bound on Q at large c that scaled like c. There is an intriguing possibility however that the true bound scales like c1=2, and that this is obscured because it takes more and more derivatives to reach this optimal bound as c increases. The basic idea for why one might expect a c1=2 scaling is that in higher dimensions, the scaling of the WGC limit can easily be read o by demanding that the binding energy from gravity and a Coulomb force cancel each other out. In the AdS3 case, one can think of the binding energy from gravity as 3 2 Q2; demanding equality would set Q pc=3, i.e. 8GN m Q 6:9c 1=2.8 c , whereas from a k = 1 Chern-Simons gauge eld exchange it is For comparison, in gure 2 we have also shown a trend line at c 1=2, which becomes further below our best numeric bound with nD = 29 derivatives as c increases. We can try 8One can read o the coe cients by looking at the vacuum conformal block for Virasoro and Kac-Moody algebras in the limit z 1.38475 1.31208 ● data of c = 105 ● ● ● ● ● ● 20 ● ● ● ● ● ● ●● 50 of derivatives used in the semide nite programming analysis increases, for the speci c case c = 105. The bound value is still changing rapidly at nD = 41. nD to estimate the optimal bound by taking our result at each c as a function of nD and extrapolating to nD = 1. The \nD = 1" points we show in gure 2 t our results starting at 15 in order to get the extrapolation. However, there is signi cant uncertainty in the resulting estimate, as can be gauged by the fact that performing the t starting at smaller or larger values of nD gives di erent answers. In gure 3 we have shown the bound as a function of nD, where one can explicitly see that the bound is still changing rapidly as a function of nD even at the upper range of what we have been able to achieve numerically. In gure 2, the error bars we have shown indicate the range over all the di erent positive values we obtain if we perform the t over nD 11; nD 13; : : : nD 19. With better numerical accuracy at large values of c, it should be possible to more rmly establish this scaling behavior. 4.4 Bound on lowest charge Next we will focus on bounds on the lowest charge Q of all charged states in the theory. We will rst consider the charge Q only, and then see how to do better by including information on dimensions and spins. To determine the upper bound on the gap to the smallest jQj of all the charged states, we want to nd a linear functional satisfying the following conditions: And jQj Q (4.11) The resulting bound on Q is shown in gure 4. The result is somewhat surprisingly always just Q = 1. This may be because in some theories with c 1 a state of Q = 1 saturates the bound and theories of larger c can be constructed as a direct product of such theories and other algebras. (vac) near the kink in the left plot is magni ed and shown in the right plot. In any case, we next turn to including information on dimensions by bounding gaps in both Q and simultaneously | Q as the lowest charge of charged states and as the lowest dimension of all non-vacuum states. To obtain such a bound, the linear functional should satisfy 50 100 ● ● ● And jQj Q (4.12) We take the linear functional to have no spin information. At each individual c, the bound carves out a region in a two-dimensional parameter space. The exclusion curve of an c = 2 example is shown in gure 5. There is a kink at 0:5 which has a bound Q 1. It would be interesting to identify what if any theory lives at this kink. Finally, the simutaneous and Q approach can be even more powerful if we turn on spin. For this case the example we choose is c = 8. In [10] the Sugawara theory E8 1.0 ●■ ◆ lattice shows up as a kink of bounds on lowest dimension of scalar primaries. We seek a And jQj And jQj Q Q (4.13) for all l 2 Z 0. The two parameter plot of and Q is shown in gure 6. Note that we nd that Q at = 0 is smaller than 1, better than the bound obtained without Q information.9 More interestingly, we see a sharp kink at = 2. In the next subsection we will analyze the extremal functional of this kink and see that this kink is the E8 lattice CFT, and we will obtain the dimension and charge spectrum of the low lying states. 4.5 Extremal functional analysis function saturating various bounds. 4.5.1 Maximal r at c = 1 In this subsection, we will use extremal functional analyses to determine the partition Our rst application of extremal function methods will be to the bound on the charge-tomass ratio r. Since our bounds have converged for c = 1 and we can consider the extremal functional for this case; by design, is non-negative on the space of states we allow, and so the states in the theory must be at the places where vanishes. The functional depends on both and Q, so the extremal spectrum contains more data as shown by gure 7. 9It is also possible that by assuming integer spins we throw away the theory that saturates the Q = 1 bound. 4 3 2 Q 1 0 c = 1, nD = 29 0 1 2 3 Δ 4 5 Q 12r c this line in gure 8. , so the extremal spectrum comprises states where = 0 in this region. There is a line of small black dots where = 0 along the Q = 0 axis that are di cult to see and so we plot along The zeros of occur at points and can be di cult to see in gure 7. In gure 8, we show along two particularly relevant lines: the neutral (Q = 0) states, and the states that saturate the mass-to-charge ratio, i.e. Q = 12r In section 4.4 we found a kink in the simultaneous and Q approach with spin information at c = 8. In order to show that the kink is indeed the level 1 E8 lattice, we can look for extremal avored partition functions by using a \two-step" approach where we rst solve for the spectrum of dimensions and then solve for the spectrum of charges. The idea is that we can use the extremal functional method on the un avored partition function, maximizing the gap in dimensions of operators. This step is just the standard extremal functional method and it will just reproduce previous results [10]. Then, having xed the weights of the states in the theory, we can impose a gap in the charge of the states in the theory, allowing only the weights (h; h) found previously. For concreteness, we will focus on the case c = 8 as a representative example. As shown in [10], the gap in dimensions is maximized at = 2 by the E8 theory at k = 1 (this theory can be described as 8 free bosons on an E8 lattice), and the extremal functional method alc = 1, nD = 29, Q = 0 Q = 12r*Δ/c = 1.41419 Δ best bound on 8GN m is shown on the right. l = 0 l = 1 l = 2 l = 3 4 Δ 2 6 8 lows one to independently derive the partition function of this theory. We have reproduced the extremal functionals ` ( ) at each spin ` in gure 9, which implies the spectrum is ( = 2; 4; 6; : : : ; l = 0; = l; l + 2; l + 4; : : : ; l 6= 0: (4.14) Moving on to the spectrum of charges, we nd that the gap Q is maximized at 2 Q = p1 . The corresponding extremal functionals ;`(Q) at each dimension and spin ` are plotted in gure 10. We note that this is not the avored partition function that one obtains if one turns on a chemical potential for the charge J = @ in the E8 lattice description; that choice corresponds to the spectrum of charges 12 Z rather than 1 1 1 1 1 1 ` 1 1 1 -1 -1 -1 jQj jQj d ;`;Q 0 1 p p 2 2 0 0 0 0 0 0 0 1 p p 2 2 134 112 2 134 112 2 1 1 1 1 1 1 ` 1 1 1 -1 -1 -1 jQj jQj d ;`;Q 0 1 2 1 0 0 0 0 0 0 0 1 2 1 92 128 28 92 128 28 at Q 2 p12 Z, whereas the right assumes Q 2 21 Z. p12 Z. Instead, if one chooses one of the length-2 vectors ~ in the E8 lattice, then J V~ 1 p 2 e~ ~ + e ~ ~ has k = 1, and the lowest charged states include for instance V2~ , with charge p1 . One can see in gure 10 that the extremal functional has zeros at around 0; p22 for all ; `, and we expect that this would continue to be true at pn2 for all n 2 Z as the number nD of derivatives used in the analysis approaches in nity. Because these dimensions and charges appear to follow such a simple pattern, we will proceed by assuming this pattern continues. Then, with the allowed weights ; ` and charges Q xed in the theory, solving the modular bootstrap equation reduces to a linear programming problem, which is much more e cient numerically.10 We obtain the occupation numbers indicated in table 2, where we have avored separately by both holomorphic and anti-holomorphic charges Q and Q. We show the occupation numbers of the conserved ` = 1 currents assuming the extremal charge spectrum Q = n2 (right). In addition, it is straightforward to repeat the analysis assuming Q = pn2 (left) for comparison. In both cases, we obtain a total of 248 currents each in the holomorphic and anti-holomorphic part, but distributed di erently among di erent charges in the two cases. (4.15) 2 2 p1 and 5 5.1 Non-abelian bounds Bounds on gaps in operator dimensions Next we turn to a numeric analysis of gaps in non-abelian theories. In some cases, the results are somewhat stronger or weaker depending on whether or not we allow for states that saturate the unitarity bound 2kh Q2, which we will refer to as \extremal states", and whether or not we impose gaps in all charge sectors. We will present results starting with the strongest assumptions rst. 10Furthermore, since this linear programming analysis xes the partition function for us to be a particular avoring of the E8 theory, by uniqueness it will be the correct one, justifying the original Ansatz. l,30 ( Δ o l 20 15 Δ=1 l=1 Δ=2 l=0 Δ=2 l=2 Δ=3 l=1 In all cases, we will present only the results for SU(2) gauge group at level k = 1. We have analyzed larger gauge groups and higher levels and the results are qualitatively similar, though the rate of numeric convergence is worse; some preliminary results for SU(2) with k = 2 are shown in appendix C. To begin, we will set the gap in all representations to be the same and restrict to the partition function at q = q; the resulting gap value will be the lowest dimension of the primary operators. This \uniform bound" is shown in gure 11. We have actually done two slightly di erent analysis, which are compared to each other on the right in gure 11. These analyses di er in whether or not we allow states in the nontrivial representations with h(0) = h(0) = 0, which saturate the unitarity bound in both the holomorphic and anti-holomorphic sectors and which we will refer to as \extremal states;" in the analysis where such states are included, the \gap" for each representation is de ned as the smallest (0) among the non-extremal states. As one can see, the di erence between the results of the two analyses is signi cant at small c but becomes negligible as c approaches 1. Ultimately, this result does not tell us much more than one learns from previous similar analyses without avored information; all we learn here is that there must be some state in the theory with (0) below some value, which is very similar to the bound on the same quantity from the un avored modular bootstrap. Next, however, we will turn to an analysis that maximizes the bounds separately in di erent sectors, and this is where we will start to nd something qualitatively new compared with what is possible with the un avored modular bootstrap. In particular, we will maximize the gap in the trivial representation, and not impose any constraint on the gaps in the other representations. In equations, our conditions are ( (0) (0) 0; 0 when = (0) and = (0); or 6= (0); (5.1) ◼ ▲ ◼ ▲ Left: bounds obtained with di erent values of nD when extremal states are not allowed. Dashed lines from blue to red are computed data of 3 nD 29. The solid black line is extrapolated from data of 11 nD 29 using a function linear in 1=nD. Right: bounds extrapolated to nD = 1 for the analysis without extremal states compared to the result when extremal states are allowed. The di erence is negligible at large c but signi cant at small c. 2 3 c(0) 4 5 nD = 5 representation obtained at increasing derivative order of the linear functional (from blue to red, up to nD = 29). The black curve is the extrapolated value. Right: blown-up plot of the-near minimal region. The minimal is expected at 2:00 c 2:04, 0:995. in SU(2) k = 1, weight (or ) takes values (0) or ( 12 ). The resulting bound on the neutral sector gap is shown as a function of c in gure 12.11 Notably, there is a minimum of about = 1 near c c(0) + 1 = 3. We next turn to a more detailed study of this point. Extremal functional analysis at c = 3 Spin-independent analysis Our q = q analysis in subsection (5.1) found a minimum gap bound at c = 3. Using the extremal functional method [19], the dimensions and the degeneracies of states at this point can be extracted, with numerical accuracy being best for the lowest dimension states. The dimensions of states occur at the zeros of the extremal functional, plotted in gure 13. 11By contrast with the previous subsection, here we nd that the bound is exactly the same whether or not we allow extremal (h(0) = h(0) = 0) states in the non-trivial representations. 1012 108 0 nD=41 λ, λ: {(0),(0)} {(0),( 12 )}, {( 12 ),(0)} Furthermore, we nd that the occupation numbers of the lowest-energy states of the partition function are uniquely determined to be M(0);(0)( ) 0( ) + 28 1( ) + 76 2( ) + 274 3( ) + : : : M(0);( 12 )( ) = M( 12 );(0)( ) M( 12 );( 12 )( ) 8 0:5( ) + 48 1:5( ) + : : : 8 0:5( ) + 48 1:5( ) + : : : The subscript on the analysis takes (0) denotes the non-Sugawara dimension of the state. At this point, 2i to be pure imaginary, so no information on spins is used: (0) = q 12 c ( Qn1=2(1 q (0) Qn1=1(1 qn) 2; qn) 2; (0) = 0; (0) > 0 We nd that a manifestly modular invariant partition function that reproduces this is Z( )z=z = = 1 1 2 22 + 42 + 42 44 z 2 4 z 3 2 where i i(z = 0). It is straightforward to check that the occupation numbers are non-negative, so that this partition function is unitary, modular invariant, and extremizes the scalar gap. Therefore (5.7) is the correct partition function by uniqueness. 5.2.2 Spin-dependent analysis In the previous subsection, we used extremal functional techniques to determine a unique partition function on the subspace q = q when the gap in the scalar sector was maximized for c = 3. We can gain much more information about the theory by relaxing the constraint q = q and varying q; q independently; in particular, the analysis becomes sensitive to the spins h h of the spectrum. We could continue to use semi-de nite programming methods, (5.2) (5.3) (5.4) (5.5) (5.6) ; (5.7) but they converge less quickly for independent q; q than they do for q = q. Instead, we can use the fact that we know the spectrum of dimensions from the q = q analysis, and the fact that spin is quantized. This allows us to x the allowed values of h; h to a discrete set, turning the problem into a linear programming problem and thereby making the analysis much more e cient. There is a remaining ambiguity, however, which is that we have to make a choice about what spins are allowed. We nd that if we allow only integer spins, there is no allowed partition function and in fact we can reduce the bound on the gap somewhat to about 2/3. If instead we allow fractional spins, then we nd a few di erent possibilities depending on what spins we allow. We will begin with the conventional case where we allow integer and half-integer total spins, h h. The SU(2) Sugawara characters are such that M(0);(0) only has integer spins, M(0);(1=2) and M(1=2);(0) only has quarter spins and M(1=2);(1=2) can have integer and half integer spins. Then to meet the requirement M(0);(0) and M(1=2);(1=2) can only have spins 24n and M(0);(1=2) and M(1=2);0 can only have spins 2n4+1 . Performing the linear programming analysis for such a spectrum (and continuing to maximize the gap in the neutral sector) leads to the following unique set of weights and occupation numbers d:12 HJEP02( 18 )4 ( ; ) ( ; ) ( ; ) 1 ; 1 2 2 12The reader may notice that the numbers at each dimension in eq. (5.2) do not match the total number of states in the table. The reason is that without knowledge of spin, there are null states that could not be taken into account in (5.2). For instance, at level 2, there are a total of 84 states, as compared with 76 in (5.2), because of the 8 conserved currents at spin 1 that consequently have 8 \null" descendants at = 2. At 3, the presence of such null states causes states to get reorganized in increasingly complicated ways and it is easiest to check the number of states is the same by constructing the full partition function. The non-Sugawara dimensions (0) and spins `(0) are just (0) = h(0) + h(0); `(0) = h(0). States are evenly divided between `(0) = +j`(0)j and `(0) = j`(0)j at each weight, and the occupation numbers for the ( ; ) = (0; 12 ) representations are the same as for ( 12 ; 0). To get the full characters one multiplies the non-Sugawara Virasoro characters h( ) (i.e. generated by the modes of T (0) = T T (sug)) by the Weyl characters (k)( ; z), which in this case are13 ( 1 )( ; z) = qm2 ym: 1 X m2Z+ After some trial and error, we nd that the corresponding avored partition function partition function is half integrally modded, it is a little unfamiliar. A natural guess is that it arises as a Z2 orbifold of a fully modular invariant theory.14 Indeed it is possible to project this onto a fully modular invariant partition function. Taking the un avored expression for simplicity, Z(inv)( ; ) = 1 2 (Z( ; ; 0; 0) + Z( + 1; + 1; 0; 0) + Z( 1=( + 1); 1=( + 1); 0; 0)) Unfortunately, we have not been able to identify a theory corresponding to (5.10). It is straightforward to check by exhausting the possibilities that the central charge and the number of spin-1 conserved currents (11) is not consistent with this partition function being associated with a pure Sugawara theory for some Lie algebra. While other choices for the quantization of spin are less conventional, they are still of some interest.15 Another possibility we have considered is that the non-Sugawara part of the spin, i.e. h(0) h(0), is an integer or a half-integer. Because of the contribution to the weight from the Sugawara part of the stress tensor, in this case the states in the ( 12 ; 0) and (0; 12 ) representations have quarter-integer spins. Performing the linear programming analysis making this Ansatz for the spins, we nd not just a unique solution but in fact a family of solutions given by the following occupation numbers: (5.8) (5.9) (5.10) ( ; ) d 1 16 12 36 32 20 ( ; ) d 4 + x 24 + 6x 4 24 4 x 6x x ( ; ) 1 ; 1 2 2 d 2x 2x 12x 2x 48 12x 13See eg [12], eqs (14.176) and (15.244). 14There is actually a history of extremal theories arising in such a fashion [28{30]. 15For instance [31]. 50 ■ 40 y 10 4 0 ●▲ ●▲ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●■ ●■ ● ● ▲ ▲ ■ ■ degeneracy of χ1/4 χ1/4 at M (1/2),(0) 8 ● χ1 χ1/2 at M(1/2),(1/2) ■ χ5/4 χ1/4 at M(1/2),(1/2) ▲ χ3/2 χ0 at M(1/2),(1/2) nds a range of possible partition functions if we allow the physical spins to take quarter integer values. When we x one of the degeneracies d by hand, in this case that of the weight ( ; ; h(0); h(0)) = ( 1 ; 0; 14 ; 14 ), all other degeneracies become 2 uniquely determined, so that we nd a one-parameter family of solutions. The degeneracy on the x-axis here is 4 + x in the notation used in the text. These partition functions satisfy crossing for all x. This one-parameter family is shown graphically in gure 14, where we perform the linear programming analysis with the degeneracy d of the ( ; ) = ( 12 ; 0); (h(0); h(0)) = ( 14 ; 14 ) chosen by hand and look at how several other degeneracies depend on this choice. By inspection of the above table, demanding that all occupation numbers be non-negative integers imposes x 2 f0; 1; : : : ; 4g. It would be interesting to know if all or any of these partition functions correspond to underlying physical CFTs. The di erent values of x here correspond to partition functions that have the same spectrum of dimensions = h + h, but which can be distinguished by their representation content, i.e. through the \ avored" partition function.16 5.3 Constraints on representation content In this nal subsection, we will consider the question of what representations are forced to be present in a theory. The gravitational AdS dual of any such constraints would imply that even if certain representations were not present among the perturbative degrees of freedom in some theory, they would have to be present non-perturbatively. The strongest condition one might try to prove is that all theories have all representations present. This would however be too ambitious since there are simple counter-examples, but one might still try to prove restrictive constraints on which representations can be absent. We will only be able to take a very modest step in this direction and prove some simple results 16They are similar in this respect to multiple di erent CFTs at c = 24 that have the same spectrum but di erent underlying symmetries [32]. for SU(2). For instance, without referring to numerical methods, we will prove that an SU(2); k = 1 partition function either has all representations, or else its avored partition function factorizes into a Sugawara theory partition function times a non- avored partition function, assuming left-right symmetry. We begin by proving this k = 1 result. The avored partition function splits into four representations M(0);(0)( ; ) M( 12 );(0)( ; ) S = p = SM ( ; )S ; = SM ( ; )S ; where now the transformation matrix is S(l)(l0) = r 2 k + 2 sin k + 2 (l + 1)(l0 + 1) : Because there are only two (assuming ( 12 ; 0) and (0; 12 ) are symmetric) di erent nontrivial representations, and the modular transformation manifestly forces at least one to be In this case, the (1; 2) entry of matrix equation (5.12) is present, we can delete only one of them. What if we set M( 12 );(0)( ; ) = M(0);( 12 )( ; ) = 0? 1 2 M(0);(0)( ; ) M( 12 );( 12 )( ; ) = 0: The two diagonal representations have to be the same and therefore the \non-Sugawara" dependence of the avored partition function is just an overall avor-independent prefactor M(0);(0)( ; ) that factors out. 12 M(0);(0)( ; ), so again the non-Sugawara holomorphic matrix, i.e. 21 10 ! = 10 10 ! + 11 00 !. this case, the residual \Sugawara" matrix is just a symmetric holomorphic plus antiSimilarly, if we set M( 1 ; 1 )( ; ) = 0, then the (2; 2) entry of (5.12) is M( 12 );(0)( ; ) = 2 2 -dependence factors out completely. In Beyond SU(2) k = 1, similar arguments can be used to somewhat narrow down the possible combinations of representations in any partition function with non-abelian currents. Spe cially, we can prove for SU(2) any k, the partition function factorizes into a Sugawara partition function times a avor-independent partition function if only diagonal representations are allowed. We again begin with the transformation rule (5.11) (5.12) (5.13) (5.14) (5.12) (5.15) Since we allow only diagonal representations, we can write for some arbitrary functions fl and gl. The matrix equation can be written as M(l)(r) M(l)(r)( ; ) = (l)(r)fl; 1 ; 1 = (l)(r)gl ; g S = S f : For = 0, k + 2 g sin ( + 1) = f0 sin k + 2 ( + 1) so g = f0 for all unless sin k+2 ( + 1) = 0. But the unitary bound < k + 1 does not allow this to happen. So all g should be equal. Therefore, all diagonal representations M(l)(l)( ; ) have to be equal, and the -dependence f0( ) of M ( ; ) completely factors out. 6 Discussion and future directions One of the main goals of this paper has been to demonstrate how systematic numeric bootstrap techniques can be applied to avored partition functions. We have considered several speci c analyses, but there are many more that could be done. Here we will discuss a few potential future directions. Some of the analyses we have discussed raise questions that could be answered with improved numeric e ciency so that the results could converge to the optimal bound. One such case is the bound on the charge-to-mass ratio, where improved accuracy at large c could more rmly establish the large c scaling of the bound. Another case is the application of our nonabelian extremal methods to larger k and larger symmetry groups. As either of these gets larger, the convergence rate becomes slower and so we have focused on the most e cient case, SU(2) at level k = 1, to demonstrate that here the extremal functional method can be used to determine the full partition function of the theory maximizing the gap in the neutral sector. It is interesting that the point maximizing the gap has integer occupation numbers, and it would be interesting to know if this is part of a general pattern or just an exceptional case. Our preliminary analysis of SU(2) at level k = 2 has not converged well enough to answer this question, but perhaps this would be possible with additional innovations or more computing power. Of course, if it turns out that integer occupation numbers is a generic feature of maximal gap spectra, it would be interesting to understand the underlying reason. As part of this question, one might consider whether the gap should be maximized in just the neutral sector or in several charged sectors. Having integer occupation numbers is a necessary but not su cient condition for a partition function to have an underlying CFT. Generally, it would be interesting to develop more techniques for determining a CFT once its partition function is known. One way is simply to use the regular bootstrap but restricting all dimensions to those that appear in the partition function. Usually, this is a signi cant improvement since it reduces the regular bootstrap problem to a linear programming problem; however, for rational theories, the (5.16) (5.17) (5.18) (5.19) HJEP02( 18 )4 large degeneracy at each level severely mitigates how helpful this additional information is. Another possible approach one could try would be to use the partition function formulated as the four-point function of twist operators, Z / h 2 2 2 2i, to include the partition function together with h i and the \mixed" correlator h 2 2i for some local operator . One could also try to make contact at c = 24 with the Schellekens classi cation [32] in terms of Neimeier lattices, by rederiving this constraint using only the avored modular bootstrap. The modular bootstrap alone cannot constrain the number of currents, since they simply contribute a constant to the partition function, but a constant no longer satis es the correct transformation law after avoring. Looking farther a eld, one of the main motivations for developing a proof of the transformation law (1.2) in terms of background elds was that this might be easier to generalize. There are many theories in 2d with higher spin currents, and one could generalize our derivation to such cases. The correlators of higher spin currents do not have a simple universal generating functional like spin-one currents do, but their correlators are severely constrained by holomorphicity and crossing, and recursion relations are known in many cases. Potentially, one could work out the transformation rule in a case-by-case fashion. More ambitiously, one could try to generalize to d > 2. The very interesting recent work [33] on a sort of modular invariance for lens spaces in higher dimension is tantalizing from this point of view. Again, one would face the issue that correlators of currents in d > 2 are not universal, but one could nevertheless try to obtain a constraint on the partition function in terms of the data in the hJ (x1) : : : J (xn)i correlators. Somewhat more abstractly, one of the appealing features of understanding the avored partition function better is that, by turning on background elds, we are exploring constraints beyond the class of those that can be seen by inserting local operators. There are many such constraints on CFTs that are invisible in the standard bootstrap; the partition function itself can be though of as one such generalization since mapping to the torus (equivalently, inserting twist operators 2) involves imposing new boundary conditions, and adding background elds is another kind of generalization. It would be very interesting to understand what additional constraints could be obtained by imposing crossing symmetry of correlators in the presence of background elds. Understanding the transformation law (1.2) as a statement about crossing symmetry for the four-point function h 2 2 2 2i would be a useful warm-up case and could potentially give insight into how to think about more general correlators. Acknowledgments We thank Scott Collier, Shamit Kachru, Jared Kaplan, Emanuel Katz, and Per Kraus for useful conversations. ED was supported in part by the National Science Foundation under grant NSF-PHY-1316699 and by the Stanford Institute for Theoretical Physics and in part by the Simons Collaboration Grant 488655 on the Non-Perturbative Bootstrap. ALF and YX were supported in part by the US Department of Energy O ce of Science under Award Number DE-SC-0010025, and ALF was supported in part by the Simons Collaboration Grant on the Non-Perturbative Bootstrap. are all the elds of the CFT, and to keep track of the torus before and after conformal transformations we have introduced a; b for two of its corners (i.e. the four corners are at 0; a; b, and a + b). Under rescalings, the operators O and parameters HJEP02( 18 )4 ( a; b) ! ( a0 ; b0) = ( a; O(w; w) ! O0(w; w) = Z a; b and consequently h Z a; b In particular, for a conserved current J , we have d e S a; b [ ] 2i R a; b dwdwAwJw = d 0e S a0; b0 [ 0] 2i R a0; b0 dwdw 1AwJw : Integrating both sides obtains the relation (Aw; a; b) = ( 1Aw; To obtain the transformation under U : +1 , we take Path integral modular transformation In this appendix, we review how di eomorphism invariance and rigid rescalings imply the relation We begin with invariance of the path integral measure: ZPI a + b cz (A.1) (A.2) (A.4) (A.5) (A.6) (A.7) (A.8) dwdwJ w(w) = dwdw J 0w( w) = dwdwJ 0w(w): a0; b0 ( a; b) = ( ; + 1) = ( ; 1); Aw = i z : z0 = 2iIm( 0)A0w = z + 1 : where the congruence = follows from a large di eomorphism cutting the torus along the line from 1 to + 1 and sewing it back to the line from + 1 to + 2. By inspection of the chemical potential term 21 i R ;1 dwdwAwJ w = 2 iIm( )AwJ0, we read o that Finally, we take = ( + 1) 1, so ( a0 ; b0) = ( +1 ; 1) and A0w = ( + 1)Aw. Therefore, The transformation under T : + 1 is trivial, since and + 1 are related by a large di eomorphism without any need for a rescaling, so = 1, and neither Aw nor z transform. All other modular transformations are generated from T and U . A \systematic" treatment to multivariate problems The bootstrap of avored partition function introduces another continuous quantum numbers Q in addition to the scaling dimension of . Unlike the un avored bootstrap where the problem is rigorously converted to a semide nite programming problem, bootstrap problems with more than one variables do not have a simple and rigorous conversion to semidefinite programming problems. One can choose to discretize the second variable Q and hope that the bound converges at very small Q. However, the bound obtained in this way is not rigorous. The linear functional can be negative in between discrete Q's or at large enough Q. B.1 Multivariate positive de nite functionals Whether any real positive semide nite polynomials (PSD) can be written as sum of squares of real polynomials (SOS) is known as the Hilbert's 17th problem. Hilbert himself proves the special case for univariate polynomials is true. But for multivariate polynomials it is later proven that PSD is a sum of squares of real rational functions. We do not like rational functions because we have much less numerical control over them than polynomials. Although we cannot nd a clean SOS representation of multivariate PSD, if we only consider the subset of strictly positive polynomials we can still represent them by SOS in the following cases: Workaround 1: multiply by a common denominator. p(x1; x2) is positive de nite polynomial (PD, also denote as p(x1; x2) > 0) then pg(x1; x2) = (1 + x21 + x22)gp(x1; x2) is a sum of square of polynomial (SOS) for some g. [34] Workaround 2: region is bounded. For a compact region S de ned by fi(~x) 0 over set of function fi, any polynomial strictly positive in S can be written as the following form p = X sI (~x)fi1 fi2 : : : I (B.1) where sI (~x) are sum of squares. I denotes some combinations of fi's. [35] The hope is that PD can approximate PSD well enough so that in practise we can still resort to SOS. Numerically, solvers like SDPB never give nonnegative polynomials with exact zeros, so in practise we never actually encounter any counterexamples. Another reason to be hopeful is from the proof that PSD can be approximated as closely as desired by SOS [36]. There is a possible loophole | the positive region of the polynomial has to be bounded. In un avored case the region that is frequently used is ?, which is a rare special case of unbounded region. In practise, there is risk of not covering the full space of PD. Although not rigorously, one can hope that by multiplying the (1 + x21 + x2)g factors of 2 higher and higher g we lose less and less. B.2 Multivariate problems and SDPB In this subsection we discuss how to rewrite the semide nite polynomial programming with 2 variables into a form suitable for SDPB [37] solver. SDPB solves univariate \Polynomial n j n maximize y0 + X bnyn n such that pj0(x1; x2) + X ynpjn(x1; x2) 0 for all x1 0 and all x2 and 1 j Matrix Program" (PMP) question stated as follows: maximize y0 + X bnyn n such that Mj0(x) + X ynMjn(x) 0 for all x 0 and 1 each single variables QA(x1; x2) = QA1(x1) QB(x1; x2) = QB1(x1) Q2(x2) Q2(x2) where M matrices are symmetric matrices of polynomials of x. Pn ynMjn(x) 0 if and only if In SDPB, the PMP question is internally mapped to an SDP question since Mj0(x) + Mj0(x) + X ynMjn(x) = tr [YAQA(x)] + x tr [YBQB(x)] for some YA; YB We are instead trying to solve the problem for two variable cases. Here for modular bootstrap we are in the special case where the symmetric matrices Mj are one by one, in other words, single polynomials pj . For simplicity here we only deal with this one dimensional case. Generalization to more dimensions and more variables is very easy. The question is stated as follows: Since SDPB only allows one variable to be bounded we cannot add more constraints on the variables. The x1 > 0 is needed in SDPB because we usually choose the input The second variable can be the U( 1 ) charge Q, which is not constrained to be positive number. If one does want to bound the second variable one can make change of variable. We use the symbol Fj to represent the linear functional Fj (x1; x2) pj0(x1; x2) + X pjn(x1; x2)yn n Similar to the univariate case, we assume that Fj 0 is equivalent to nding YA;j ; YB;j 0, so that Fj (x1; x2) = tr [YA;j QA(x1; x2)] + x1tr [YB;j QB(x1; x2)] Here we introduce \bilinear basis" ~q(X) so that Q(X) = ~q~qT spans the space of polynomials of X. An easy example of bilinear basis is ~q(x) = f1; x; x2; : : :g. The bilinear basis of two or more variables can be factored out as a kronecker product of bilinear bases of (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) Since a polynomial is xed if we know its value at (d + 1) di erent points, we can simply evaluate the above equation at (d2 + 1) values of x2 in order to reduce the equation to have only one variable x1 Fj;k(x1) = Fj(x1; x2;k) = tr YA;j QA1(x1) Q2(x2;k) + x1tr YB;j QB1(x1) Q2(x2;k) The above (d2 + 1) equations are equivalent to (B.9). In the following we omit the j index because the same equation works for all j. Now the form is already in single variable and is very close to the form of (B.3). The only di erence is the numerical matrices Q2(x2;k). Here we can play a trick by shu ing the (d2 + 1) equations with linear combination klFl = tr YA;j QA1(x1) + (B part) (B.11) X l X l for some dimension (d2 + 1) square matrix kl. In fact, the space of symmetric Q2 matrices is only (d2+1 = 2 2 1) dimensional space since it spans the space (d2+1) dimensional polynomials. That means we can always nd some kl which picks up the orthornormal basis of the polynomial space. Further we can perform an arbitrary GL 2 transformation on Q2 so that the orthornormal basis maps to the symmetric matrix basis X l l klQ2(x2;l) 7! G 1 X klQ2(x2;l)G G 1 X l klQ2(x2;l)G = Er(k)s(k) We de ne di to be the xi degree of the polynomial F . the dimensions of the matrices Q are dimQA1 = A1 = [d1=2] + 1 dimQB1 = B1 = [(d1 After factoring out QA and QB the function Fj is written as Fj(x1; x2) = tr YA;j QA1(x1) Q2(x2) + x1tr YB;j QB1(x1) Q2(x2) where Ers = ir js + jr is. Then X l h klFl = tr YAQ1A(x1) Er(k)s(k)i + x1tr hYBQ1B(x1) Er(k)s(k)i Compared to (B.3), we can turn double variable programming of polynomial into single variable programming of symmetric polynomial matrices by substitution k l k l Mj0 = X X Er(k)s(k) klPj0(x1; x2;l) Mjn = X X Er(k)s(k) klPjn(x1; x2;l) (B.8) (B.9) (B.10) (B.12) (B.13) (B.14) (B.15) * Δ = ( ) to the lightest neutral state for SU(2) at level k = 2. The bound is minimized at c 2:715. extremal functional analysis with SU(2) at k = 2 as a function of c. The optimal bound is at c 2:715, indicated by a vertical line; horizontal lines are shown at integers. Since Q2 span a (d2 + 1 = 2 2 1) dimension space it means only the diagonal and next-to-diagonal elements will be nonzero. If further we only have even powers of x2, the matrices will be diagonal. C k = 2; SU(2) analysis Here we present some preliminary results on our methods applied to the group SU(2) at level k = 2. Our results are qualitatively similar to the k = 1 case, though with worse numeric accuracy due to the slower convergence. In gure 15, we show the bound on the gap to the lightest neutral state in the theory, which is minimized to be 1:344 at c 2:715. Unfortunately, at the point where the bound is minimized, the occupation numbers from our analysis for some of the lowest few states are not particularly close to integers. It is not clear whether this indicates that such a point is not associated with an underlying CFT or if we simply have not converged to su cient precision. The occupation numbers for the lightest neutral state and charged state are shown as a function of c in gure 16. The lightest neutral state is close to d = 74, however the lightest charged state, which is even lighter is relatively far from the nearest integer, d ought to maximize the gap in not only the neutral sector but also in one or more charged sectors; it would be interesting to pursue this or other conditions further. Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [2] S. Hellerman, A universal inequality for CFT and quantum gravity, JHEP 08 (2011) 130 Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE]. [arXiv:0902.2790] [INSPIRE]. theory, JHEP 08 (2011) 127 [arXiv:1007.0756] [INSPIRE]. 180 [arXiv:1307.6562] [INSPIRE]. (2008) 193 [hep-th/0609074] [INSPIRE]. [3] S. Hellerman and C. Schmidt-Colinet, Bounds for state degeneracies in 2D conformal eld [4] D. Friedan and C.A. Keller, Constraints on 2D CFT partition functions, JHEP 10 (2013) [5] P. Kraus, Lectures on black holes and the AdS3/CFT2 correspondence, Lect. Notes Phys. 755 [6] N. Benjamin, E. Dyer, A.L. Fitzpatrick and S. Kachru, Universal bounds on charged states in 2d CFT and 3d gravity, JHEP 08 (2016) 041 [arXiv:1603.09745] [INSPIRE]. [7] R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, C = 1 conformal eld theories on Riemann surfaces, Commun. Math. Phys. 115 (1988) 649 [INSPIRE]. [8] R. Dijkgraaf, G.W. Moore, E.P. Verlinde and H.L. Verlinde, Elliptic genera of symmetric products and second quantized strings, Commun. Math. Phys. 185 (1997) 197 [hep-th/9608096] [INSPIRE]. [9] M. Krauel and G. Mason, Vertex operator algebras and weak Jacobi forms, Int. J. Math. 23 [10] S. Collier, Y.-H. Lin and X. Yin, Modular bootstrap revisited, arXiv:1608.06241 [INSPIRE]. [11] J.-B. Bae, S. Lee and J. Song, Modular constraints on conformal eld theories with currents, JHEP 12 (2017) 045 [arXiv:1708.08815] [INSPIRE]. [12] P. Di Francesco, P. Mathieu and D. Senechal, Conformal eld theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York U.S.A., (1997) [INSPIRE]. [13] J.D. Qualls and A.D. Shapere, Bounds on operator dimensions in 2D conformal eld theories, JHEP 05 (2014) 091 [arXiv:1312.0038] [INSPIRE]. [14] J.D. Qualls, Universal bounds on operator dimensions in general 2D conformal eld theories, arXiv:1508.00548 [INSPIRE]. [15] S. Collier, P. Kravchuk, Y.-H. Lin and X. Yin, Bootstrapping the spectral function: on the uniqueness of Liouville and the universality of BTZ, arXiv:1702.00423 [INSPIRE]. [16] M. Montero, G. Shiu and P. Soler, The weak gravity conjecture in three dimensions, JHEP 10 (2016) 159 [arXiv:1606.08438] [INSPIRE]. JHEP 08 (2017) 025 [arXiv:1606.08437] [INSPIRE]. [17] B. Heidenreich, M. Reece and T. Rudelius, Evidence for a sublattice weak gravity conjecture, functional method, Phys. Rev. Lett. 111 (2013) 241601 [arXiv:1211.2810] [INSPIRE]. Institute in Elementary Particle Physics: new frontiers in [arXiv:1602.07982] [INSPIRE]. 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE]. model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE]. non-unitary bootstrap, JHEP 11 (2016) 030 [arXiv:1606.07458] [INSPIRE]. from the CFT bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE]. from classical background elds, JHEP 11 (2015) 200 [arXiv:1501.05315] [INSPIRE]. [arXiv:1602.06930] [INSPIRE]. conformal quantum eld theory and selfdual critical points in Zn invariant statistical systems, Sov. Phys. JETP 62 (1985) 215 [Zh. Eksp. Teor. Fiz. 89 (1985) 380] [INSPIRE]. S3 and circle [1] O. Lunin and S.D. Mathur , Correlation functions for M N =SN orbifolds , Commun. Math. [18] D. Das , S. Datta and S. Pal , Charged structure constants from modularity , JHEP 11 ( 2017 ) [20] S. Rychkov , EPFL lectures on conformal eld theory in D 3 dimensions , SpringerBriefs Phys . ( 2016 ) [arXiv: 1601 .05000] [INSPIRE]. [21] D. Simmons-Du n , The conformal bootstrap , in Proceedings, Theoretical Advanced Study elds and strings (TASI 2015 ), [22] R. Rattazzi , V.S. Rychkov , E. Tonni and A. Vichi , Bounding scalar operator dimensions in [23] F. Kos , D. Poland and D. Simmons-Du n , Bootstrapping mixed correlators in the 3D Ising [24] D. Poland , D. Simmons-Du n and A. Vichi, Carving out the space of 4D CFTs, JHEP 05


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP02%282018%29148.pdf

Ethan Dyer, A. Liam Fitzpatrick, Yuan Xin. Constraints on flavored 2d CFT partition functions, Journal of High Energy Physics, 2018, 148, DOI: 10.1007/JHEP02(2018)148